In a few words, the structure of this nonlinearity originates from the probabilistic construction of multifractal random fields using the a Gaussian multiplicative chaos [1]}, [2]}, [3]}. In our setup, to be more precise, and as it is demanded by the complex nature of our dynamics (Eq. ), we will invoke a complex generalization of this probabilistic object, some aspects of which have been already explored in the literature [4]}. It consists in introducing a random field able to reproduce key ingredients that enter in the nonlinear behavior of the spectrum of exponents \(\zeta _q\)  of high-order structure functions (Eq. REF ). It is obtained as the exponential of a logarithmically correlated Gaussian random field. It can be seen as a particular case of the more general class of log-infinitely divisible measures [5]}, [6]}, [7]}, [8]}, [9]}, [3]} and has been extensively used under various forms while modeling the random nature of fluid turbulence [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}. Moreover, the Gaussian field entering in the construction, that we recall to be logarithmically correlated, can be seen, in a way that we will discuss later, as a fractional Gaussian field of vanishing parameter \(H=0\)  [18]}. Not only is this remark important because such a field can thus be defined as a solution of regularized versions of random walks [19]}, [20]}, [21]}, but also because it makes a clear connection with the aforementioned build-up of fractional Gaussian fields using the operator \(P_H\)  (Eq. REF ) for the boundary case \(H=0\) . For several reasons that are developed in the sequel during the ad-hoc construction of multifractal fields, it turns out necessary to introduce a Hermitian symmetric version \(\widetilde{P}_0\)  of the Fourier multiplier of such an operator \(P_0\)  (Eq. REF ) that reads
\(\widetilde{P}_0u(t,x) \equiv -i \int e^{2i\pi k x}\frac{k}{|k|_{1/L}^{3/2}}\widehat{u}(t,k)dk.\) 
